The first initial-boundary value problem for some nonlinear time degenerate parabolic equations

Abstract:

Consider the nonuniformly parabolic operator \[ Lu = \sum \limits _{i,j = 1}^n {{a^{ij}}(x,t){u_{{x_i}{x_j}}} + \sum \limits _{i = 1}^n {{b^i}(x,t){u_{{x_i}}} - c(x,t,u){u_t} + d(x,t)u,} } \] where u, ${a^{ij}},{b^i}$ c, d are bounded, real-valued functions defined on a domain $D = \Omega \times [0,T] \subset {R^{n + 1}}$. Assume that $c(x,t,u)$ is Lipschitz continuous in $|\bar \cdot |_\alpha ^D$ of ${C_\alpha }(D)$, and that $c(x,t,u) \geqq 0$ on D. Sufficient conditions on c are found which guarantee existence of a unique solution $u \in {\bar C_{2 + \alpha }}$ to the first initial-boundary value problem $Lu = f(x,t), u = \psi$, on the normal boundary of D, where $\psi \in {\bar C_{2 + \alpha }}$. Existence is proved by direct application of a fixed point theorem due to Schauder using existence of a solution to the linear problem as well as a priori estimates.